MATH128 Lecture Notes - Lecture 1: Ibm System P, Ratio Test
21 Feb 2016
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0 cos2 (x) + sin2 (x) = 1. 1 + cos (2x) cos2 (x) = sin2 (x) = 2 arn is convergent if |r| < 1 and is divergent if |r| 1 given that a 6= 0. The divergence test: if lim n an does not exist or lim n an 6= 0 then the series. The integral test: suppose f (x) is a continuous, positive, eventually decreasing function on [1, ) such that f (n) = an then. 1 np converges when p > 1 and diverges when p 1. Xn=0 bn are series with positive terms. (a) if (b) if. Xn=0 bn converges and an bn for all n then bn diverges and if an bn for all n then. Alternating series test: if the alternating series ( 1)nbn (where bn 0 for all n) satis es the two conditions lim n bn = 0 and bn+1 bn for all n then the series converges.
